Math Problem Statement
f(x)= x−2 4x−5 ,what is the equation of asympote
Solution
To find the equations of the asymptotes for the given rational function:
Step 1: Vertical Asymptote
The vertical asymptotes occur when the denominator equals zero (and the numerator does not also equal zero at the same point). Set the denominator and solve for :
So, the vertical asymptote is:
Step 2: Horizontal Asymptote
For horizontal asymptotes, we compare the degrees of the numerator and denominator:
- The degree of the numerator () is 1.
- The degree of the denominator () is 1.
When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficients are:
Thus, the horizontal asymptote is:
Final Answer
The equations of the asymptotes are:
- Vertical asymptote:
- Horizontal asymptote:
Would you like a detailed graph of this function or additional explanation?
Related Questions:
- How do you determine the oblique asymptote of a rational function if applicable?
- Can a function have more than one vertical asymptote?
- What happens to as for this specific function?
- How do horizontal asymptotes differ from slant (oblique) asymptotes?
- Can the graph of intersect its horizontal asymptote?
Tip:
When identifying asymptotes, always check for cancellations in the rational function to ensure there are no "holes" that might affect the vertical asymptote.
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Asymptotes
Algebra
Formulas
Vertical Asymptote: Set denominator equal to zero and solve for x.
Horizontal Asymptote: If degrees of numerator and denominator are equal, asymptote is the ratio of leading coefficients.
Theorems
Vertical Asymptote Theorem
Horizontal Asymptote Theorem
Suitable Grade Level
Grades 9-11
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